The Wonderful World ofAftershocks

Karen FelzerUSGS

Aftershocks are often seen as small,unimportant earthquakes

• The National Hazard Map andengineering codes ignore them!

• But aftershocks are important!* Majority of all earthquakes* Release tectonic stress* Amazing statistical properties* Mysterious physics….* The key to the propagation of tectonic

earthquakes ?

Omori’s Law: Aftershock rate decays intime as

!

R(t) =K

(t + c)p

• Aftershocks can follow Omori’s law for hundreds ofyears (Ebel et al., 2000).

• Omori’s law is empirical. No one has been able toderive it.

Amazing aftershock statistic #1

• Larger earthquakes have larger aftershocks onlybecause they have more aftershocks.

• All foreshocks can be explained as earthquakes withaftershocks that are larger than themselves.

Amazing aftershock statistic #2

The magnitude of each individualaftershock is independent of the

magnitude of its mainshock[Reasenberg and Jones (1989), Michael and Jones (1998),Felzer et al. (2004)].

Evidence for Statistic #2: Aftershockmagnitude distributions are independent ofmainshock magnitude

Aftershocks taken from 2 days/5 km around each mainshock.3200 aftershocks in each distribution

Gutenberg-Richterdistribution

Foreshock rates can be accuratelypredicted from the rate of aftershocks

smaller than the mainshock1. No. aftershocks ≥ M = F(Mmain)10-bM

2. F(Mmain) ~ 10Mmain - 1.3 in California, for aftershocks within1 day and 1 fault length of mainshock; b=1.

GR distribution

Predicted rate that an earthquake will produce anaftershock larger than itself within one day:

=10Mmain-1.3-M = 10Mmain-1.3-Mmain = 0.050

Observed rate: 0.047 +- 0.0054(6086 M ≥ 3 California earthquakes, 1984-2004)

Amazing aftershock statistic #3

The distance of an aftershock from itsmainshock is independent of the

magnitude of the mainshock[Felzer and Brodsky, 2006]

• Distant aftershocks of small earthquakes aresimply difficult to see because there are so fewaftershocks/mainshock

• The trick to seeing the distant aftershocks is tocombine lots of aftershock sequences

Spatial stack of the first 30 minutes of M≥2aftershocks of 2,355 M 3--4 mainshocks

Mainshocks

M 3.5 fault length ~ 0.6 km

199 aftershocksplotted

The distribution of aftershock distances isindependent of mainshock magnitude

Aftershocks taken from within 30 minutes of the mainshocks. 200aftershocks in each data set

The rate of decay of aftershock densityis also consistent over long distances

Aftershock density decay with distance is constant from 0.05to 280 Fault Lengths => Dynamic triggering throughout

(Felzer and Brodsky, 2005)

Amazing aftershock statistic #4

The timing or ‘clock advance’ of anaftershock is independent of the amount of

stress applied by the mainshock[Felzer 2005]

This observation negates most physicalaftershock generation models, includingthe static stress change + rate and state

friction of Dieterich (1994) !

In the static stress + rate and state model,aftershocks are earthquakes strongly time

advanced by an amount proportional tothe mainshock-applied stress

It’s like a mousenudging clock

hands -- theamount of clock

advance ∝ amountof nudge

“Mouse nudge” model expectation:Lower stress change at larger distances ∝

later aftershocks

Aftershock densitydecreases with

distance/stress change

And the mean time ofaftershock occurrence

increases withdistance/stress change

Alternative: The “Harold Lloyd” ModelBinary, not proportional, stress response

Option 1: Stiff clock or lightHarold => no clock advance

Option 2: The hand gives waymaximum clock advance

If Harold is heavier, a clock advance is more likely

But Harold’s weight ≠ clock advance size

Harold Lloyd model expectation:Distribution of aftershock times is the

same at all distances

Aftershock density stilldecreases withdistance/stress

But the mean time ofaftershock occurrencedoesn’t change withdistance/stress

Data: Mean aftershock time does not varywith stress change

Averageaftershock time,0-33 km

Averageexpected forrandomquakes

Aftershocks measured over first 2 days of sequences, M 5-6 mainshocks

M 7.3 Landers earthquake: Average aftershocktime is independent of stress change

Averageaftershock time

Averageexpected forrandomquakes

Aftershocks measured over first 30 days of sequence

Another test of Harold vs. the mouse:Inspect full seismicity decay rate

“Mouse nudge” predictionFewer rapid aftershocks atlower stress: Omori Law c

value increases with distance

“Harold Lloyd” predictionDecay rate the same at all stress

changes

Based on Fig 2,Dieterich, 1994

M 5 - 6 mainshocks: Seismicity rate changedecay does not depend on distance

Agrees with the results of Jones and Hauksson (1998) that the valueof c is constant with distance

Landers mainshock: Groups of 200 aftershocks atdifferent distances show the same decay rate

KolmogorovSmirnoff Test:All distributionssimilar at 95%confidence

Result: The correct physicalaftershock model must contain binary,

on/off aftershock triggering

• Many stochastic aftershock models(Kagan (1982), Ogata (1998), and others)use binary triggering.

• But I am not aware of a physicalaftershock model that satisfies binarytriggering ?

• New physical models needed!

Amazing aftershock statistic #5

Static stress changes appear inefficient attriggering aftershocks

• Pollitz and Johnston (2006) found that for similarearthquake and aseismic events, the earthquakesproduced many more aftershocks.

• Reliable stress shadows cannot be found (Marsan, 2003;Mallman and Parsons 2004; Felzer and Brodsky 2005).

• Static stress change patterns improve aftershocklocation predictions by ~8% (Parsons, 2002). Dynamicstress patterns do at least as well (Kilb et al. 2000).

Amazing aftershock statistic #6

If you project Omori’s law back far enough,you can make a mainshock!

If the time gap betweenevents comes from here,we see a single, larger

earthquake

If the time gap comes fromhere we see a separate

mainshock and aftershock

Kagan and Knopoff (1981)

Statistic #6 suggests that mainshock areavalanches of subevent aftershocks.

Subevent earthquake models are common• Housner (1955)• Vere-Jones (1976)• Kagan and Knopoff (1981)• Zeng et al. (1994)• Rydelek and Sacks (1996)• Lavallee and Archuleta (2003)• …….

The innovation is to connect the “avalanche”& aftershock processes

Conceptual Model

Earthquake remains small An aftershock subevent istriggered; earthquake grows

• A subevent nucleates at a point of weakness

• The subevent is stopped by strength heterogeneitieswhen it reaches area A unless its co-seismic stress cantrigger additional subevents, via aftershock-trig physics.

• Each subevent has the same average area (controlled bythe heterogeneity) and potential to trigger others

Fault plane

Points ofhigh

strength

OR

Preliminary stochastic Monte Carlomodeling shows that the aftershock

avalanche model can produceearthquakes with Gutenberg-Richter

magnitude-frequency stats andrealistic time series

Monte Carlo Simulation Basics

• Model starts with a few random subevents• Locations of triggered subevents: Chosen randomly from distribution cr-1.4

(Felzer and Brodsky, 2006)

• Time of triggered subevents: S wave travel time

+ minimum wait time (tmin) + Omori’s Law (at-p)• Subevent triggering is binary. Timing is not

dependent on stress change, and events canonly be triggered once.

Monte Carlo Simulation Parameters

Each subeventproduces ~ 1

aftershock0.007 secondstmin

ReproducesBåth’s Law for> 20 s bet. eqs

0.0013a

Felzer et al.(2003)1.34p

ReasonValueParameter

Monte Carlo Simulation: Combiningsubevents into earthquakes

Δt

Unit subevent

Composite earthquake

• If Δt < (0.2)( no. subevents in existing earthquake)) =>Subevent is added to the preceding earthquake

• Composite Magnitude= log(Earthquake Area)

timefudgefactor Proxy for earthquake duration

Simulation results: G-R relationship, b = 1

Simulation results: Time series ofcoalesced earthquakes

Direct observation of the proposedsubevents: High frequency bursts

following Omori’s Law during Chi-Chi

Chen et al. (2006) and Fischer and Sammis (2005)

The aftershock avalanche modelagrees with our observed aftershock

statisticsObserved:

Aftershock magnitude, location, and timing areindependent of mainshock magnitude

Model:Predicts observation. Aftershock triggering isaccomplished by the initiation of tiny subeventsthat are the same in all earthquakes => allaftershock properties should be independent ofmainshock magnitude.

Agreement between model andobserved aftershock statistics

Observed:The time an aftershock occurs isindependent of the amount of stress applied.

Model:This observation is one of the rules in themodel. Without it, earthquakes wouldaccelerate and scale invariance would not bepreserved.

Agreement between model andobserved aftershock statistics

Observed:Most or all aftershocks are most likelytriggered by dynamic stress changes

Model:Aftershock triggering and mainshockpropagation are the same thing, and mainshockpropagation is understood to be a dynamicprocess.

Conclusions• Aftershock magnitude, distance, and timing

are independent of mainshock magnitude.• Aftershock timing is independent of

mainshock-induced stress => aftershockscannot be triggered by static stress change +rate and state friction.

• The idea that earthquake propagation =aftershock triggering is consistent withearthquake & aftershock statistics.